[EM] Ratings as a standard

[Bart Ingles]

Blake Cretney wrote:
> But, if we elected B in the following example, wouldn't that be "an
> election whose winner has an extremely low rating, when other
> candidates have much higher ratings."  Even if you abandon average
> ratings as a standard, but still require that methods not be too
> different from it, you run into the same problem.
> 
> 1  voter   A 500 B 0
> 40 voters  A 0   B 2

[BI]
If this is an open-ended scale, as you seem to intend, then it is
meaningless as far as I'm concerned.  It's not something I would
consider using, much less defending.  There is no way to enforce any
sort of equal voting power, unless you allow voters to use plus or minus
infinity as a value.  And then how do you indicate a choice midway
between infinity and zero?

If this is a 0-500 scale, then the 40 voters don't appear to support
either candidate to any meaningful degree.  Such a candidate field is
the result of a bad nomination process, or else this is a partial
candidate field -- possibly a pairwise total in a larger Condorcet
election, or perhaps this is a runoff election.  Either way I wouldn't
consider the result to be meaningful using either ratings or rankings.

 
> > Again, here is the kind of example I was concerned with:
> >
> >      Voter utility
> >      100 99.9 99.8                                    0.2 0.1 0.0
> >      ---------------------/\/--      --/\/-----------------------
> > 499   A                                                      B  C
> > 3     B  A                                                      C
> > 498   C                                                      B  A
> >
> > It seems to me that your other examples are irrelevant.  If your
> > hypothesis is that the Condorcet winner is always desirable, and I give
> > an example like the one above, your choices are to either explain why B
> > is a better candidate than A _in_this_example_ or else reject your
> > hypothesis.

[BC]
> I assume that voters have a slightly greater probability of backing
> correct propositions than incorrect ones.  It follows that B is most
> likely better than A because more voters said this than said the
> opposite, and there are no contradictory majority decisions.

[BI]
I take it that you also assume that this "slightly greater probability"
is equal regardless of the degree of preference?  If not, then your
conclusion does not follow.  You don't explain why a nearly indifferent
voter is as likely to be right as a moderately concerned voter. 
Wouldn't an indifferent voter be easily swayed by frivolous information,
or by strategy? 

[BC]
> Note that the only reason given for rejecting B in this situation is
> that B scores too low on average ratings.  I have never expected that
> you would reject a ratings result simply because it was too different
> from Condorcet, but you expect me to reject a Condorcet result simply
> because it is too different from ratings.

[BI]
Average ratings is not really the point -- my reason for rejecting B is
that almost all voters consider B to be almost as bad as the worst
possible choice, while an actual majority of voters consider A to be
acceptable.

It has been argued that pairwise methods make no assumptions about
strengths of preferences between pairings, but in fact pairwise systems
do assume that all such pairings are equally relevant, as you indicate
above.

Given only rankings, the only assumptions we can safely make about a
voters' preferences are that the voter substantially prefers his first
choice over his last choice, otherwise he would not bother to vote
(unless you make voting mandatory).  There is no way to know what a
ranking between first and last signifies, if anything.  In other words,
we don't know whether the voter strongly approves or strongly
disapproves of a middle candidate, or something in between.  Where the
middle candidate stands in relation to the known first and last choice
is arguably more important than where that candidate stands in relation
to another middle choice of unknown significance.


> > If you contend that the outcome is a result of extremist voting, then
> > which voters are the extremists?  Since only three voters strongly favor
> > B, they would seem to be the most suspect.  Does this mean that
> > Condorcet favors the extreme voters in this case?  If not, then since
> > they also favor A, how could the 499 A voters be considered extremist?
> 
> Consider a vote on how much money will be spent on an upcoming
> project.  Voters have a range of opinion between $0 and $1000, and
> favour amounts closest to their first choice over amounts further
> away.  Let's assume that the median view is $500, which is represented
> by candidate B, and supported only by 1 voter.  A will represent a
> lower amount, C a greater amount.
> 
> Now, B's plurality support is largely dependent on the placement of A
> and C.  If A and C represent $499.99 and $500.01, then B will only
> receive one vote.  If A and C are chosen for $0 and $1000, B will
> likely receive lots of plurality support.  This is why I do not agree
> with the argument, "Since only three voters strongly favor B, they
> would seem to be the most suspect."
> 
> Now, even though the A-1st and C-1st voters sincerely rate their
> favourite at a great distance from the next candidate (B), this may
> simply be a result of a personality trait.  The B-1st voters may be
> more willing to compromise than either the A-1st or the C-1st.  Of
> course this doesn't mean that any of them are extremists, in the sense
> of the word you intend.

[BI]
I don't know what you're trying to show here, or how it relates my
question.  In your scenario, there is only one policy dimension, and all
candidates are equidistant.  Since B is the median voter, he is by
definition not an extremist.  This is different from the example on
which my question was based.  Whether B receives lots of plurality
support in your example has no bearing on whether there is an extremist
in mine.  And why would they vote differently with 499.99/500.01 than
with 0/1000?  Under what voting system?


Looking back at my example, it occurs to me that if it were a real
pairwise election, the results would tend to follow average ratings
anyway, rather than the sincere CW.  Even with the best polls, 499:498
is a statistical dead heat.  If the A and C voters both truncate, they
each have a .5 probability of winning, so the utility to each of these
voters is 0.5 * 100 = 50 (on a scale of 0-100).  If they vote sincerely,
they guarantee B's win with a utility of only 1.0 * 0.1 = 0.1.

In fact it looks as though the A and C voters should truncate whenever
they would bullet vote under approval voting, at least under a
symmetrical 'bad CW' example where cycles are avoided.  The main
difference from approval voting is that under pairwise voting, a naive
voter might dutifully rank all candidates even though this is not in his
own best interest.  Another difference is that if half of the A and C
groups rank sincerely, it is possible for the other half of the two
groups to nullify the sincere voters' choices, rejecting the CW even
when Approval might have selected the CW.


> What do you mean then by talking about "absolute" ratings.

[BI]
Take a universe of potential candidates, containing all candidates who
are eligible and willing to run under reasonable circumstances, and are
the favorite of at least one voter.  You can peg each voter's favorite
at 100, and his least favorite at zero.

"Absolute" may not be the correct word, since this scale is relative to
the best and worst potential candidate for each voter, but it is
certainly not relative in the sense that it depends on who is actually
in the race.  You could probably say this scale is "relative" if
interpreted as Von Neumann-Morganstern utilities, but it is not relative
to some arbitrary subset of candidates.

If the nominating process is fair and provides a field of candidates
representative of the "ideal" field of candidates, then most voters
should have an actual candidate who rates reasonably close to 100, and a
candidate rated reasonably close to zero.  If the field is sufficiently
representative, you should be able to use 100 and 0 as approximations of
absolute ratings.

Turning this around, if you have a good nominating process, then you
should be able to use 100 and zero as the range for actual candidates. 
If this is off too far, the nominating process is at fault; if off only
a little, the actual ratings should still be more representative of
"ideal" ratings than rankings would be.